The appeal came the day after the president, complying with the law, signed an order bringing arbitrary cuts worth $85 billion into force as well as a report by his Office of Management and Budget (OMB) detailing the cuts to each agency.
Obama has called the sweeping cuts, stemming from a 2011 debt ceiling agreement, "dumb."
The across-the-board cuts were triggered automatically following the failure of efforts to clinch a deal with Republicans on cutting the deficit.
But in his weekly radio and internet address, he argued there was still time to find a smarter solution to the nation's debt problem.
"I still believe we can and must replace these cuts with a balanced approach - one that combines smart spending cuts with entitlement reform and changes to our tax code that make it more fair for families and businesses without raising anyone's tax rates," Obama said.
He said the budget deficit now exceeding $1 trillion can be reduced without laying off workers or forcing parents and students to pay the price.
"A majority of the American people agree with me on this approach - including a majority of Republicans," the president argued. "We just need Republicans in Congress to catch up with their own party and the rest of the country."
Under the sequester, 800,000 civilian employees of the Defense Department will go on a mandatory furlough one day a week and the navy will trim voyages. The deployment of a second aircraft carrier to the Gulf has been canceled.
Defense contractors may be forced to lay off workers and some federal health spending could be hit.
Cuts will also be made to special needs education and preschool for less well-off children. National parks could close and wait times could hit four hours at airport customs posts.
But the president insisted that despite public bickering, Republicans and Democrats actually had more in common than they were willing to let on.
"I know there are Republicans in Congress who would actually rather see tax loopholes closed than let these cuts go through," Obama said. "And I know there are Democrats who'd rather do smart entitlement reform than let these cuts go through. There's a caucus of common sense. And I'm going to keep reaching out to them to fix this for good."
Obama had earlier blamed the austerity time bomb on Republicans, who refused to close tax loopholes for the rich and corporations combined with more targeted spending cuts, in his "balanced" approach to deficit reduction.
"I am not a dictator. I'm the president," Obama said Friday, warning he could not force his Republican foes to "do the right thing," or make the Secret Service barricade Republicans leaders in a room until a deal is done.
"These cuts will hurt our economy, will cost us jobs and to set it right both sides need to be able to compromise," Obama said, before decrying the budget trimming as "dumb" and "unnecessary."
Only three months after winning re-election, and with the extent of his authority in Washington again constrained, Obama bemoaned his inability to do a "Jedi mind-meld" to get Republicans to change their minds, using imagery from Star Wars and Star Trek.
Obama was bound by law to initiate the automatic, indiscriminate cuts by the end of Friday.
The hit to military and domestic spending, known as the sequester, was never supposed to happen, but was rather a device seen as so punishing that rival lawmakers would be forced to find a better compromise to cut the deficit.
Both sides agree that the sequester is a blunt instrument to cut spending, as it does not distinguish between essential and wasteful programs -- in what Obama has branded a "meat-cleaver" approach.
New Defense Secretary Chuck Hagel warned that the sequester could endanger the military's capacity to conduct its missions.
"Let me make it clear that this uncertainty puts at risk our ability to effectively all of our missions," said Hagel. The Pentagon's budget is set to be slashed by roughly $46 billion.
In the report to government agencies, the OMB said non-exempt defense programs would be cut by 13 percent this year and domestic programs would be sliced by nine percent.
The president met with chief political foes -- Republican House Speaker John Boehner and Senate Minority Leader Mitch McConnell -- and allies Democratic Senate Majority Leader Harry Reid and top House Democrat Nancy Pelosi in the Oval Office Friday.
But Boehner emerged from the talks to signal to reporters that Republicans would not budge on Obama's key demand for a deal that would raise tax revenues.
"Let's make it clear that the president got his tax hikes on January 1. This discussion about revenue, in my view, is over," Boehner said.